Slow Transverse Mode Research Articles

Transverse and Longitudinal Degradations in Ceramic Solid Electrolytes

Ceramic solid electrolytes conducting primarily a specific ion but with minuscule electron leakage are used in many electrochemical devices. Their degradation phenomena may be classified into two broad categories─a slow transverse mode accumulating damage normal to the ionic current direction and a fast longitudinal mode accumulating degradation parallel to the current direction. Examples of the transverse mode include oxygen bubbles on transverse grain boundaries, in-plane electrolyte cracking in solid oxide electrolysis cells, and cathodic reduction of beta-alumina electrolytes in Na–S batteries. Examples of the longitudinal mode include short-circuiting dendrite formation in metal batteries of both the Na–S type and the all-solid-state lithium metal type. Analogous instability modes are also seen in field-assisted ceramic processing, dielectric devices, and memristors. These phenomenological similarities across different devices, operating conditions, and technologies, as well as the origins of their damage mechanisms, can be understood in terms of the highly nonlinear spatial distributions of the minority carriers (electron/hole) and the chemical potential of the equivalent charge-neutral species such as O2; such distributions can in turn cause or exacerbate damaging concentrations of stress and electric fields. This review also outlines strategies for improving material designs to mitigate degradation, which will be especially important for operations and applications under extreme electrochemical conditions.

Effect of Phonon Focusing on the Drag Thermopower in Single-Crystal Potassium Nanowires at Low Temperatures

The role of shear waves and the effect of phonon focusing on the anisotropy of the thermopower of the electron–phonon drag in single-crystal potassium nanowires at low temperatures are analyzed in this work. The deformation potential theory is used for the longitudinal components of elastic modes. For the shear components of vibrational modes, the electron–phonon interaction constant is used, which was determined previously from comparing the results of calculating the thermoelectric power with experimental data for bulk potassium crystals. It is shown that shear waves make a significant contribution to the drag thermopower of nanowires. In the regime of the Knudsen phonon gas flow for nanowires with a cross-section side D = 5 × 10–6 cm, the contribution of the slow transverse mode t2 with allowance for only the longitudinal component in the [111] directions turned out to be 32% smaller and, with allowance for the shear component of the t2 mode, 12% greater than the contribution of longitudinal phonons. Directions corresponding to the maximum and minimum values of the drag thermopower of nanowires are determined. It is shown that, under conditions of competition between the boundary and bulk mechanisms of phonon relaxation with an increase in the cross section of nanowires, the anisotropy of the drag thermopower changes nonmonotonically. It exceeds 30% not only in the Knudsen phonon gas flow regime, but also when the sample thickness is two orders of magnitude greater. This makes the anisotropy of thermopower available for experimental studies.

The effect of phonon focusing on the electron-phonon relaxation and electron transport in potassium crystals

The influence of anisotropy of elastic energy on electron-phonon relaxation and the role of shear waves in the electrical resistance of potassium crystals are investigated. It is shown that, at temperatures much lower than the Debye temperature (T<< θD), the contribution of slow quasi-transverse phonons to the electrical resistance of potassium crystals exceeds that of longitudinal phonons by an order of magnitude. Earlier, the Bloch-Grüneisen theory left aside this component under the above conditions. At the same time, at high temperatures(T>>θD), the contribution of longitudinal phonons to the electrical resistance turns out to be 4 times greater than the total contribution of electron relaxation by fast and slow transverse modes. The role of shear waves in the electrical resistance of potassium crystals is analyzed. It is shown that, at low temperatures, this mechanism provides 32% of the total electrical resistance. It is 4 times higher than the contribution of longitudinal phonons to the electrical resistance and should be taken into account when analyzing the electrical resistance of alkali metals. The distribution function of the most effective phonons for electrical resistance is defined, and the inelasticity of electron-phonon scattering is analyzed. It is shown that the calculated results of the electrical resistance of potassium in the temperature range from 40 to 400 K, taking into account the anisotropy of elastic energy, are in good agreement with the experimental data without the use of fitting parameters.

Elastic Properties, Phonon Focusing and Electronic Structures of Typical Long-Period Superstructures Al&lt;sub&gt;5&lt;/sub&gt;Ti&lt;sub&gt;2&lt;/sub&gt; and Al&lt;sub&gt;11&lt;/sub&gt;Ti&lt;sub&gt;5&lt;/sub&gt;

The first-principles calculations were performed to systematically study the elastic properties, phonon focusing and electronic structures of typical long-period superstructures Al5Ti2 and Al11Ti5. The obtained lattice parameters are in good agreement with the experimental data. The elastic constants were calculated, and bulk modulus B, shear modulus G, Young’s modulus E and Poison’s ration ν were further obtained. Both of the numerical indicators and the three dimensional images show that Al5Ti2 and Al11Ti5 are anisotropic, and the anisotropy is slightly high for Al5Ti2. Three slowness surfaces were constructed for Al5Ti2 and Al11Ti5 to investigate the phonon focusing patterns, and the group velocity surfaces were also obtained to gain more insight into the elastic anisotropy. Due to elastic anisotropy, both of the slowness surfaces and the group velocity surfaces are nonspherical, and the anisotropy is highest for slow transverse mode, and then followed by fast transverse mode and longitudinal mode. The electronic density of states and charge density distribution of Al5Ti2 and Al11Ti5 indicate that due to strong hybridization between Al-2p and Ti-3d, there is a strong directional bonding between the Al and Ti atoms. Electronic structures reveal the underling mechanism of elastic properties and phonon focusing for Al5Ti2 and Al11Ti5.

Effect of phonon focusing on Knudsen flow of phonon gas in single-crystal nanowires made of spintronics materials

Effect of anisotropy of elastic energy on the phonon propagation in single-crystal nanowires made of Fe, Cu, MgO, InSb, and GaAs materials that are used to fabricate spintronics devices in the regime of the Knudsen flow of phonon gas has been studied. A new method of analyzing the focusing of quasi-transverse modes has been suggested, which made it possible to determine the average values of the densities of phonon states in the regions of focusing and defocusing slow and fast quasi-transverse modes. The effect of phonon focusing on the anisotropy of heat conductivity and lengths of the phonon free paths has been analyzed for all acoustic modes that exist in spintronics nanostructures. It has been shown that for all the nanowires investigated the angular dependences of the free paths of fast and slow transverse modes in the {100} and {110} planes correlate with the angular dependences of the densities of phonon states for these modes. Directions of the heat flux that ensure the maximum and minimum phonon heat conductivity in the nanowires have been determined.

Elastic anisotropy and phonon focusing for tetragonal crystals: Application to γ-TiAl

A theoretical formalism to investigate the elastic anisotropy and phonon focusing for tetragonal crystals is described in detail in this paper. This is applied for γ-TiAl and both of the numerical indicators and the three dimensional images of elastic anisotropy are derived. The percentage of bulk and shear anisotropies indicate γ-TiAl exhibits relatively high degree of anisotropy in shear but small anisotropy in compressibility. Anisotropy indexes A∗ and AU demonstrate γ-TiAl possesses high degree of anisotropy, and this feature is also reflected in the three dimensional Young’s modulus, shear modulus and Poisson’s ratio. Three slowness surfaces are constructed for γ-TiAl to investigate the phonon focusing patterns, and the group velocity surfaces are also obtained to gain more insight into the elastic anisotropy. Due to elastic anisotropy, both of the slowness surfaces and the group velocity surfaces are nonspherical. The anisotropy is highest for slow transverse mode, and then followed by fast transverse mode and longitudinal mode. Interestingly, only transverse phonons cause caustics for γ-TiAl. Furthermore, the phase velocity and group velocity are not collinear for transverse mode in some directions, resulting in cuspidal edges along these directions in the group velocity surfaces.

Effect of phonon focusing on the temperature dependence of thermal conductivity of silicon

The effect of phonon focusing on the temperature dependence of thermal conductivity of single crystalline silicon is studied at temperatures from 3 to 40 K in the frame of generalized Callaway theory. Thermal conductivity has been calculated for silicon samples in the form of finite length rectangular rods with different orientations of long axis and side faces. To evaluate the phonon scattering rate due to diffuse boundary scattering, the analytical expressions derived recently by us were utilized. Theoretical results representing the dependence of on temperature and sample orientation are in agreement with experimental data. Contributions of different phonon polarizations to the thermal conductivity have been analyzed. The transverse phonon branches are most important as expected, the slow transverse mode being the most prominent in the formation of conductivity anisotropy. The phonon dispersion was found to have a weak effect on the calculated at these temperatures.

Ultrasonic investigations of the Jahn–Teller effect in a ZnSe:Fe2+ crystal

The temperature dependence of the absorption and velocity of transverse ultrasonic waves propagating in the [110] direction in a ZnSe:Fe2+ crystal with the iron impurity of n=2.2×1019cm−3 is investigated in the temperature interval 1.4–100K at a frequency of 53MHz. For the slow transverse mode an absorption peak at 7.3K and a decrease of the phase velocity below 40K are found, while no such features are observed for the fast mode. The anomalies are interpreted as manifestations of the Jahn–Teller effect. The softening of the elastic modulus (C11–C12)∕2 indicates that the local distortions in ZnSe:Fe2+ are tetragonal. Under the assumption that the absorption maximum is of a relaxational nature, the temperature dependence of the relaxation time is recovered from the data, and the values of the potential barrier height and vibronic frequency are obtained.

On interference effects in the elastodynamic Green’s functions G33( x, ω) of Si and GaAs

In this paper, we analyze interference effects present in the elastodynamic Green’s functions G 33( x, ω) of the cubic crystals Si and GaAs, which are associated with folded portions of the wave surface of the slow transverse (ST) acoustic mode. G 33( x, ω) represents the three dimensional extension of the amplitude distribution imaged in the transmission acoustic microscopy of these crystals. The intensity contrast for oscillations of a particular wave vector k in the interference pattern is determined essentially by the 3D Fourier transform of G 33 ( x , ω ) G 33 ∗ ( x , ω ) . According to the Fourier autocorrelation theorem, that transform is equivalent to the autocorrelation function of the corresponding distribution G 33( k, ω) in k-space. We show that due to the linear mapping between k-space and the slowness vector s-space, the interference phenomena discussed here are related to geometrical features of the slowness surface of the ST mode. We present calculations of these effects based on the angular spectrum technique.

Ballistic phonon‐drag imaging in AlAs quantum wells

The phonon-drag voltage induced by acoustic phonons has been studied for the first time in AlAs quantum wells. The 15 nm wells are embedded in Al0.45Ga0.55As buffer layers grown on (001) GaAs wafers. Electrons in the wells couple strongly to phonons due to the large effective mass, leading to signals that are several orders of magnitude larger than those observed in similar experiments with GaAs heterostructures. Measurements of the drag voltage as a function of phonon source position (phonon-drag images) yield information about the polarization dependence of the electron-phonon coupling. The data is compared to numerical simulations including deformation potential and piezoelectric coupling. The experimental images show distinctive signals from slow transverse and longitudinal phonon modes that indicate strong deformation potential coupling. Additional modeling is needed to understand the source of the clear sign inversion of the signal near the center of the experimental images. Future studies at 3He temperatures will investigate quantum effects. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

High-pressure Brillouin study on the orientationally disordered phase of methane

Three elastic constants in the orientationally disordered solid phase of methane have been determined at room temperature, as a function of pressure up to 5 GPa, by Brillouin spectroscopy developed for single crystal grown in a diamond anvil cell. It is found that the velocity of the slow transverse acoustic mode in the 〈110〉 direction is insensitive to pressure as compared with other velocities, which leads to the anomalous increasing of the ratio of the longitudinal acoustic velocity to the slow transverse one in the 〈110〉 direction. This result suggests that the rotation-translation coupling has strong pressure dependence.

Phenomenological electrodynamics of electronic superlattices

We set up the phenomenological electrodynamics and establish the generalized dispersion relation for a model type-I superlattice. The presence of a constant externally applied magnetic field is assumed. Field equations and the fluctuation-dissipation theorem are formulated exclusively in terms of Fourier-transformed monolayer source and field quantities. The significance of the correct dispersion relation is especially striking when applied to the calculation of the eigenfrequencies of the slow transverse modes (such as the helicon).

Numerical results and Biot theory in anisotropic porous/fibrous media

Biot theory [1] is a basic tool when studying the propagation of elastic waves in porous/fibrous solids. Due to the simplicity of the assumptions, isotropic materials are most often utilized [2,3]. However, in some cases, anisotropy has to be included in the modelization, e.g. with some fibrous materials. In this work, we have developped routines, based on the standard Biot model, which compute the various characteristic surfaces (slowness, phase and group velocities) [4] in a given plane for anisotropic porous media. By increasing the weight of the coupling constants, one can continuously vary from the limit case of anisotropic solids with no porosity to a number of different porous media. When the coupling terms are not equal to zero, one finds four instead of three propagating modes, the two other eigenvalues being complex, meaning in turns that the two remaining solutions do correspond to some evanescent modes. The slow transverse mode is unchanged. The fourth mode often called the Biot mode [5], which is very similar to the pseudo-thermal wave [6,7] observed in dynamic thermoelasticity, has always a very small wavespeed and is very highly damped

Determination of all elastic constants of transversely isotropic media with a cusp around the symmetric axis by use of elastic pulses propagating in two principal directions.

This paper describes a method of determining all five elastic constants of a hexagonal or transversely isotropic medium of type II which has the cuspidal region of group velocity around the highestsymmetry axis [001] perpendicular to the transversely isotropic basal plane. The elastic constants are obtained from wave-speed data of broad-band ultrasonic signals propagating in two principal directions, namely the [001] and the other parallel to the basal plane. The method uses specific cuspidal features of the slow transverse (ST) mode propagating in the [001] direction with two distinct group velocities. An analytic formula that relates the slower group velocity of the ST mode with the elastic constant C 13 is derived

Focusing of fast transverse modes in silicon and slow transverse modes in zinc

This paper presents focusing of fast transverse (FT) modes in silicon and slow transverse (ST) modes in zinc, both observed at utrasonic frequencies. Using a (001)-oriented silicon disk, the FT modes are generated and detected by two tiny shear PZT transducers, both polarized in the [100] direction. One transducer fixed on the bottom surface acts as a source and the other serves as a detector scanning the top surface along the [100] direction across various points on the [010] axis at some distances away from the epicenter. The observed focusing pattern indicates strong concentration of FT modes on the narrow band about the (100) plane containing the source. For a (001)-oriented zinc crystal, strong focusing of ST or quasi-transverse (QT) modes toward the symmetry direction is observed: first in theoretical and experimental, epicentral Green’s functions; second with very small PZT source and PZT detector; finally with focused pulsed laser excitation and small PZT detector. The focusing patterns observed both in silicon and in zinc are consistent with the results predicted by the theory of phonon focusing. [Work supported by ONR.]

The distribution and focusing of phonons for the rare-gas solids xenon and helium

The phonon distribution and focusing of the heaviest and the lightest of the rare-gas solids, namely xenon and 3He, have been studied in detail. On the basis of the theoretical study, it is found that the phonon magnification factor (PMF) has unusually high values, of the order of 105, for the longitudinal mode for xenon. The slow shear mode also has high values for the PMF. The distribution in direction of the phonons in the group velocity space has been investigated using the Schmidt equal-area net. For the fast shear mode for xenon, the phonons are channelled along the principal symmetry planes and the phonons show very sharp focusing; there are also large islands bereft of any phonons. Since the values of C11 and C12 are very nearly equal for solid 3He, it shows an elastic anisotropy that is more similar to that of the A15 compounds exhibiting a soft mode than that of the other rare-gas solids. For the slow transverse mode, the elastic anisotropy is such that the group velocity and phase velocity vectors are nearly perpendicular. The polar Schmidt net for 3He does not show very sharp focusing traits or islands.

Phonon scattering by dislocations in LiF crystals

Abstract A theory is presented of the interaction between thermal phonons and a vibrating dislocation. An expression for the relaxation time of the phonon is obtained which is valid in an anisotropic medium. The relaxation time depends on the strength of the imaginary part of the effective mass of the vibrating dislocation. We have calculated numerically this factor for LiF and have found the mode dependence to be very strong. The slow transverse-mode component is about ten times larger than the other mode components. Hence, the slow transverse mode is scattered more strongly than other modes by vibrating dislocations in LiF. Agreement with experimental data is obtained.

Magnetoacoustic resonance in antiferromagnetic insulators in ‘‘moderate’’ and strong magnetic fields

The magnetoacoustic resonance between phonons and magnons is studied analytically on the basis of a set of linearized magnetoelastic equations, which was previously deduced from a rotationally invariant nonlinear theory for uniaxial antiferromagnets of the easy-axis type. More specifically, the present paper investigates the less studied case where the bias magnetic field is either ‘‘moderate’’ or strong relative to some material parameter. Various types of dispersion and magnetoacoustic resonance couplings are exhibited depending on whether waves propagate along the direction of or orthogonal to the symmetry axis. In some cases a fast transverse elastic mode couples with both upper and lower magnon branches, while longitudinal and slow transverse elastic modes travel independently. In another case both fast and slow transverse elastic modes couple with both upper and lower magnon branches. In the case that most closely resembles the case of ferromagnets, left-circularly polarized transverse phonons and magnons couple at both high and low magnetoacoustic resonance points, while the right-circularly polarized transverse phonons are only rendered slightly dispersive for small wave numbers. These coupled dispersion behaviors are studied analytically (asymptotics) in terms of a small parameter which represents globally magnetoelastic couplings. This paper concludes the study of magnetoacoustic resonance in antiferromagnetic insulators which was started in a previous paper concerned with the case of weak bias magnetic fields.

Electric Field Coupling to Slow Elastic Modes in Gels of Conjugated Polymers

The interaction of low-frequency ac electric fields with the anisotropic polarizability of rodlike polydiacetylene macromolecules is used to excite slow (50 cm/sec) transverse elastic modes in gels made up of a network of these molecules. The specific coupling of the ac field excitation and the birefringence detection to the network allows direct measurements of the fraction of the sample for which there is an elastic restoring force (the "infinite cluster").

Pressure dependence of the elastic moduli of semimetallic(Ti1−xVx)2O3at 296 K. Softening of a slow shear mode and two other elastic eigenvalues whenx≈0.04

Ultrasonic 30-MHz-wave travel times have been measured at 296 K for semimetallic, single-crystal ${({\mathrm{Ti}}_{1\ensuremath{-}x}{\mathrm{V}}_{x})}_{2}{\mathrm{O}}_{3}$ samples with $0.02\ensuremath{\le}x\ensuremath{\le}0.09$ as a function of hydrostatic pressure up to 4 kbar. The moduli of all waves except the slow transverse (ST) mode propagating in the [100] direction in the $x=0.04$ sample stiffen with increasing pressure. This ST mode is a symmetry-breaking one, and its modulus, ${\mathrm{C}}_{\mathrm{ST}}$, is an eigenvalue of the elastic constant matrix. Linear extrapolation of ${\mathrm{C}}_{\mathrm{ST}}$ implies a structural transition at very high pressure (1000 kbar using the Born criterion ${\mathrm{C}}_{\mathrm{ST}}$=0, and 275 kbar using a Demarest et al.-type criterion $\frac{{\mathrm{C}}_{\mathrm{ST}}}{B}=0.2$, where $B$ is the bulk modulus). The pressure dependences of all six independent elastic constants, ${C}_{11}$, ${C}_{12}$, ${C}_{13}$, ${C}_{14}$, ${C}_{33}$, and ${C}_{44}$ are determined for samples with $x=0.02$ and $0.04$ or $0.05$ and of all ${C}_{\mathrm{ij}}$'s except ${C}_{13}$ for $x=0.09$ samples. Comparison with data for semiconducting ${\mathrm{Ti}}_{2}$${\mathrm{O}}_{3}$ does not reveal any simple correlation between the $\frac{{\ensuremath{\Delta}C}_{\mathrm{ij}}}{\ensuremath{\Delta}P}$ values with electrical resistivity, lattice parameters, interatomic spacings, or the values of the ${C}_{\mathrm{ij}}$'s. It is found that the other eigenvalues of the ${C}_{\mathrm{ij}}$ matrix usually increase with pressure. Samples with $x=0.04$ or $0.05$ are exceptional again in that two other eigenvalues are decreased by pressure. One of them is associated with a symmetry-breaking mode and is approximately equal to ${C}_{11}\ensuremath{-}{C}_{12}$ because ${C}_{14}$ is very small.